Harnack inequality in doubling quasi metric spaces and applications.

In this thesis we present an axiomatic approach to an invariant Harnack inequality for non homogeneous PDEs in the setting of doubling quasi-metric spaces. We adapt the abstract procedure developed by Di Fazio, Gutiérrez and Lanconelli, for homogeneous PDEs taking into account the right hand side of the equation. In particular we adapt the notions of double ball property and critical density property: these notions arise from Krylov-Safonov technique for uniformly elliptic operators and they imply Harnack inequality. Then we apply the axiomatic procedure to subelliptic equations in non divergence form involving Grushin vector fields and to X-elliptic operators in divergence form